Question

for each problem, find the: x and y intercepts, x - coordinates of the critical points, open intervals where the function is increasing and decreasing, x - coordinates of the inflection points, open intervals where the function is concave up and concave down, and relative minima and maxima. using this information, sketch the graph of the function.

1. ( y = -\frac{x^{3}}{3}+x^{2} )

Explanation:

Step1: Find x-intercepts

Set $y=0$: $-\frac{x^3}{3} + x^2 = 0 \Rightarrow x^2(-\frac{x}{3} + 1) = 0 \Rightarrow x=0$ or $x=3$

Step2: Find y-intercept

Set $x=0$: $y=0$

Step3: Find critical points

First derivative: $y' = -x^2 + 2x$
Set $y'=0$: $-x(x-2)=0 \Rightarrow x=0$ or $x=2$

Step4: Determine increasing/decreasing

Test intervals:
$(-\infty,0)$: $y'(-1)=-3<0$ (decreasing)
$(0,2)$: $y'(1)=1>0$ (increasing)
$(2,\infty)$: $y'(3)=-3<0$ (decreasing)

Step5: Find relative extrema

At $x=0$: decreasing→increasing (minima)
At $x=2$: increasing→decreasing (maxima)

Step6: Find inflection points

Second derivative: $y''=-2x + 2$
Set $y''=0$: $-2x+2=0 \Rightarrow x=1$

Step7: Determine concavity

Test intervals:
$(-\infty,1)$: $y''(0)=2>0$ (concave up)
$(1,\infty)$: $y''(2)=-2<0$ (concave down)

Answer:

x-intercepts: 0, 3; y-intercept: 0; critical points: 0, 2; decreasing on $(-\infty,0)\cup(2,\infty)$, increasing on $(0,2)$; relative minima at $x=0$, relative maxima at $x=2$; inflection point at $x=1$; concave up on $(-\infty,1)$, concave down on $(1,\infty)$
(Sketch: Plot intercepts (0,0),(3,0); minima at (0,0), maxima at (2, 4/3); inflection at (1, 2/3); curve decreasing then increasing then decreasing, concave up then down)